Anomalous Valley-Current Generation

• Anomalous valley-current generation is a phenomenon where a net flow of valley quantum numbers occurs in 2D materials without an accompanying charge current, primarily driven by Berry curvature effects.
• Intrinsic mechanisms arise from broken inversion symmetry that creates valley-contrasting Berry curvature, while extrinsic contributions stem from engineered scattering and disorder in systems like bilayer graphene.
• Nonlinear and dynamic techniques, including strain engineering, optical pumping, and phonon drag, enable tunable valley currents, offering promising avenues for advanced valleytronic device applications.

Anomalous valley-current generation refers to all mechanisms by which a net flow of “valley quantum number” arises in condensed-matter systems, typically in the absence of a net charge current. This effect is fundamentally connected to the valley degree of freedom in materials with multiple degenerate but inequivalent band extrema (“valleys”) in their Brillouin zone, such as the K, K′ points of hexagonal 2D materials. The physical origin of anomalous valley currents spans intrinsic Berry curvature effects, extrinsic scattering, nonlinear and dynamic mechanisms, uniquely enabled by broken symmetries, band topology, or collective ordering. A comprehensive understanding now integrates quantized and nonquantized regimes in both fermionic and bosonic systems, all-optical and phononic driving, and the impact of device geometry and disorder.

1. Fundamental Mechanisms: Berry Curvature and Topological Valley Hall Effects

The most fundamental source of anomalous valley current is the valley-contrasting Berry curvature in systems with broken inversion symmetry. In such cases, each valley (e.g., K and K′) exhibits opposite-signed Berry curvature, leading to an anomalous (transverse) velocity under an applied field. In 2D materials with a massive Dirac-like bandstructure—such as monolayer MoS₂ and WSe₂—the effective Hamiltonian near each valley τ=±1 is

$$H_τ(\mathbf{k}) = \hbar v_F ( τ k_x σ_x + k_y σ_y ) + \frac{Δ}{2} σ_z$$

resulting in a Berry curvature

$$Ω_{n,τ}(k) = -τ \frac{(ħv_F)^2 (Δ/2)}{[ (Δ/2)^2 + (ħv_F k)^2 ]^{3/2}}$$

for band n, valley τ. Application of an in-plane electric field produces opposite anomalous velocities for K and K′ carriers, generating a net transverse pure valley current (no net charge current). The total valley Hall conductivity is given by

$$σ_v = \frac{e^2}{ħ} \sum_{n=c,v} \sum_{τ=±1} \int \frac{d^2k}{(2π)^2} f_{n}(k) τ Ω_{n,τ}(k)$$

This intrinsic valley Hall effect has been robustly verified: in MoS₂ devices, room-temperature nonlocal transport has demonstrated λ_v up to 0.8 μm, and θ≈σ_v/σ_xx in the range 0.05–0.5. In the presence of inversion symmetry (e.g., multilayer MoS₂), the Berry curvature vanishes and the effect disappears (Hung et al., 2018).

Spontaneous valley polarization can be realized via broken time-reversal in ferromagnetic materials such as single-layer FeCl₂, where the effective Hamiltonian includes both exchange splitting and spin–orbit coupling, giving rise to

$ΔE = 4λ_{soc}$

with a reported splitting ΔE≈101 meV and σ_{xy}≈e²/h. This leads to a valley- and spin-polarized anomalous Hall current under in-plane electric fields (Zhaos et al., 2020).

In antiferromagnetic monolayer MnBr, although the total Berry curvature is zero by PT symmetry, a spin-layer locked Berry curvature Ω_s(𝐤) survives, yielding a “spin-layer polarized” anomalous valley Hall current; ΔE_v=21.55 meV is tunable by Hubbard U, strain, field orientation, and electric field (Wang et al., 11 Nov 2024).

2. Extrinsic and Disorder-Driven Valley Currents

Beyond intrinsic effects, valley currents can arise from valley-contrasting scattering at engineered inhomogeneities. Biased bilayer graphene quantum dots with controlled interlayer potential Δ exhibit robust valley-dependent skew scattering. The low-energy Hamiltonian in the {A₂, b₂, a₁, B₁} basis under gate control reads: $$H_τ(π) = \begin{pmatrix} U_2 & v_F τ π^\dagger & γ_1 & 0 \ v_F τ π & U_2 & 0 & 0 \ γ_1 & 0 & U_1 & v_F τ π^\dagger \ 0 & 0 & v_F τ π & U_1 \end{pmatrix}$$ The extrinsic valley Hall effect is defined by integrating valley-resolved scattering cross-sections over the propagation angles,

$$σ_v^{\rm skew} \approx n_i \frac{e^2}{h} \int_0^{2π} dθ \sin θ [ σ_K(θ) - σ_{K'}(θ) ] \propto n_i ξ^v$$

Control over dot radius R, mass Δ, and centering δ governs both magnitude and direction of skewness. When such BLG dots are embedded in monolayer graphene, edge states and boundary conditions modulate the valley splitting, but robust effects remain provided edge disorder is minimized (Solomon et al., 2021).

Disorder-driven phenomena are also evident in valley-polarized quantum anomalous Hall (VQAH) phases. Competing spin-orbit couplings break valley symmetry, and moderate short-range disorder localizes pairs of counterpropagating edge modes, leaving perfectly valley-polarized chiral channels protected by the net Chern number. The transition from QAH to VQAH is signaled by the appearance of multiple valley-winding skyrmions in real-space spin textures and Chern-number jumps (Pan et al., 2014).

3. Nonlinear and Dynamical Valley-Current Generation

A crucial frontier is the realization of nonlinear valley Hall effects (NVHE) and dynamic pumping protocols. In systems with both inversion and time-reversal symmetries, the linear valley Hall effect is symmetry-forbidden, but a second-order (nonlinear) valley Hall current arises via electric-field modification of the quantum geometry (Berry connection polarizability). For tilted massless Dirac fermions, e.g., strained graphene with Hamiltonian

$$\mathcal{H}_s(\mathbf{k}) = \hbar v_F [ s k_x σ_x + k_y σ_y ] + s \hbar v_t k_x σ_0$$

the NVHE is controlled by the tilt velocity v_t. The nonlinear response tensor reads

$χ^{NLV}_{x;yy} = - λ \frac{e^3 v_t}{2π μ^2 }$

yielding a finite second-harmonic valley current under longitudinal fields. The effect is intrinsic and remains at τ-independent, broadening applicability even to centrosymmetric, nonmagnetic crystals (Das et al., 2023).

Oscillating strain profiles induce time-dependent pseudo-electric fields, as in graphene nanobubbles with dynamic out-of-plane deformation,

$\mathbf{E}_{ps} = -\partial_t \mathbf{A}_{ps}$

Simulations demonstrate remarkable valley-current amplitudes in the sub-THz regime, with pronounced higher harmonics arising from the nonlinear relation between the oscillation and the pseudo-field. Instantaneous currents can reach up to nanoamperes through narrow leads, even though the time-averaged current vanishes; rectification or combination with valley filters can yield net dc valley-bias (Hadadi et al., 2023).

Surface acoustic waves can resonantly drive anomalous valley current in intervalley-coherent phases, with a characteristic $\omega^{-2}$ divergence at low frequency, originating from a pseudo-superfluid density linked to valley gauge-symmetry breaking. This "acoustogalvanic" response is analogous to supercurrent response in superconductors and offers a bulk probe of electronic order (Tanaka et al., 11 Dec 2025).

4. Drag-Induced, Optical, and Thermally Driven Generation

Valley currents are also generated via phonon and photon drag, in which electrons are transversely deflected by Berry curvature under a dragging force. The full valley-Hall current contains skew-scattering, side-jump, and anomalous-velocity contributions; these can partially compensate, with the total response depending on disorder and phonon/impurity parameters. The valley-Hall conductivity under static field, phonon drag, or photon drag is

$$\sigma_v = \sum \text{(skew)} + \sum \text{(coh)} + \sum \text{(anom)}$$

with distinct scaling for each process ($\xi \propto \gamma^2/E_g^2$); experimental regimes can isolate different physical contributions (Glazov et al., 2020).

Photoinduced generation exploits valley-contrasting selection rules: in WSe₂, circularly polarized light selectively excites one valley, producing a net valley population and a detectable spin-coupled valley photocurrent. This current can be modulated by external electric fields exploiting inversion symmetry breaking, enabling greater than 100×

modulation in channel current via gate voltage control. The photocurrent formula is

$$\mathbf{j} = x I E_{ex} \sin\theta \sin 2\phi\, \mathbf{e}_{\parallel}$$

where the degree of circular polarization and the perpendicular field control magnitude and direction (Yuan et al., 2014). Ultrafast all-optical schemes in biased bilayer graphene synchronize valley-selective pulses with subgap THz-field kicks, generating femtosecond pure valley currents, with the generation time determined by the gap, τ_gen ≈ ħ/Δ (Gill et al., 4 Nov 2024).

Thermally driven Nernst effects, with appropriate Berry curvature, can generate pure spin and valley currents perpendicular to an in-plane temperature gradient. The valley Nernst effect allows for dissipationless valley current, highly tunable by gating and thermal control (Yu et al., 2015).

5. Correlated, Topological, and Collective Phenomena

Collective orderings and topological phase transitions provide new routes for anomalous valley-current response. In twisted bilayer graphene aligned with hBN, a DC current under broken lattice C₃ symmetry generates a valley density imbalance via anisotropic intervalley scattering. This bias enters the Landau free energy as

$F(m;J,T) = F_0(m;T) - 2m \Delta n_0(J)$

with a discontinuous first-order switching of the valley order parameter m at a coercive critical current. This abrupt jump flips the sign of the anomalous Hall conductance, observed as hysteresis in experimental Hall resistance (Ying et al., 2021).

Quantum anomalous valley Hall phases—enabled in systems such as H-FeCl₂ or indirect exciton condensates—exhibit quantized valley Hall conductance without external magnetic fields. The formation of half-valley metals (100% valley and spin polarization) and QAVHE regimes are tracked via Chern numbers: $C_{\pm}(\tau) = \mp (\tau/2) \text{sgn}[A'(\tau)]$ The edge-state structure directly reflects C_total, with a single valley-polarized chiral edge mode in the QAVHE phase (Hu et al., 2020, Kovalev et al., 2019).

Bosonic analogs, as in quantum anomalous valley Hall effect for exciton condensates, are characterized by quantized steps in valley Hall conductivity under dynamic pumping—originating from finite Berry curvature, condensate depletion, and collective motion (Kovalev et al., 2019).

6. Edge, Interface, and Device-Level Engineering

Quantum anomalous Hall topologies in monolayer graphene with staggered sublattice potential and exchange fields create interfaces (zero-line modes) supporting robust valley-polarized 1D channels. Depending on the domain Chern numbers, interfaces can realize co-propagating or valley-selective modes, whose current partitioning can be routed, split, or filtered at designed Y-junctions. Near-unity partition and full reconfigurability are achieved by tuning magnetic and sublattice parameters (Luo, 2018).

Strain engineering induces effective valley-dependent gauge fields, giving rise to quantized valley Hall plateaus and dissipationless valley currents. The valley Hall effect and the Haldane anomalous Hall effect can be unified as distinct precession axes of an emergent large pseudospin residing in the zeroth Landau level. This mapping elucidates the topological origin of the valley Hall effect and robustness to disorder (Settnes et al., 2017, Yuan et al., 2021).

7. Prospects, Control Strategies, and Material Platforms

Anomalous valley-current generation unites a range of mechanisms across 2D materials, including TMD monolayers, graphene and bilayer graphene (gated, twisted, or strained), topological insulator surfaces, organic conductors, and antiferromagnets. Multifield tuning (electric, strain, gate, ferroelectric switching) allows real-time, nonvolatile, and energy-efficient control in valleytronic devices. Candidate platforms now span systems with spontaneous symmetry breaking, engineered mass profiles, or collective valley order, supporting phenomena from quantized topological currents to ultrafast all-optical operation (Zhaos et al., 2020, Wang et al., 11 Nov 2024, Gill et al., 4 Nov 2024).

Key open directions include dynamic control of coherence and symmetry, minutely engineered device geometries for unidirectional valley flow, and integration with spin and optical manipulations for multifunctional valley-spintronic architectures. Emergent regimes with nonlinear, bosonic, or intervalley-ordered responses further deepen the field’s topological and quantum complexity.